Short-course Compressive Sensing of Videos Venue CVPR 2012, Providence, RI, USA June 16, 2012

Short-courseCompressive Sensing of Videos

VenueCVPR 2012, Providence, RI, USA

June 16, 2012

Organizers:Richard G. BaraniukMohit GuptaAswin C. SankaranarayananAshok Veeraraghavan

Part 2: Compressive sensingMotivation, theory, recovery

• Linear inverse problems

• Sensing visual signals

• Compressive sensing– Theory– Hallmark– Recovery algorithms

• Model-based compressive sensing– Models specific to visual signals

Linear inverse problems

Linear inverse problems• Many classic problems in computer can be posed

as linear inverse problems

• Notation– Signal of interest

– Observations

– Measurement model

• Problem definition: given , recover

measurement noise

measurement matrix

Linear inverse problems

• Problem definition: given , recover

• Scenario 1

• We can invert the system of equations

• Focus more on robustness to noise via signal priors

Linear inverse problems

• Problem definition: given , recover

• Scenario 2

• Measurement matrix has a (N-M) dimensional null-space

• Solution is no longer unique

• Many interesting vision problem fall under this scenario

• Key quantity of concern: Under-sampling ratio M/N

Image super-resolution

Low resolution input/observation

128x128 pixels

Image super-resolution

2x super-resolution

Image super-resolution

4x super-resolution

Image super-resolution

Super-resolution factor D

Under-sampling factor M/N = 1/D2

General rule: The smaller the under-sampling, the more the unknowns and hence, the harder the super-resolution problem

Many other vision problems…• Affine rank minimizaiton, Matrix completion,

deconvolution, Robust PCA

• Image synthesis– Infilling, denoising, etc.

• Light transport– Reflectance fields, BRDFs, Direct global separation, Light

transport matrices

• Sensing

Sensing visual signals

High-dimensional visual signals

Human skinSub-surface scattering

Electron microscopyTomography

Reflection

Refraction

FogVolumetric scattering

The plenoptic functionCollection of all variations of light in a scene

Adelson and Bergen (91)

space (3D)

angle (2D)

spectrum(1D)

time (1D)

Different slices reveal different scene properties

space (3D)

angle (2D)

spectrum(1D)

time (1D)

Lytro light-field camera

High-speed cameras

The plenoptic function

Hyper-spectral imaging

Sensing the plenoptic function

• High-dimensional– 1000 samples/dim == 10^21 dimensional signal– Greater than all the storage in the world

• Traditional theories of sensing fail us!

Resolution trade-off

• Key enabling factor: Spatial resolution is cheap!

• Commercial cameras have 10s of megapixels

• One idea is the we trade-off spatial resolution for resolution in some other axis

Spatio-angular tradeoff

[Ng, 2005]

Spatio-angular tradeoff

[Levoy et al. 2006]

Spatio-temporal tradeoff

[Bub et al., 2010]

Stagger pixel-shutter within each exposure

Spatio-temporal tradeoff

[Bub et al., 2010]

Rearrange to get high temporal resolution video at lower spatial-resolution

Resolution trade-off

• Very powerful and simple idea

• Drawbacks– Does not extend to non-visible

spectrum• 1 Megapixel SWIR camera costs 50-

100k

– Linear and global tradeoffs

– With today’s technology, cannot obtain more than 10x for video without sacrificing spatial resolution completely

Compressive sensing

Sense by Sampling

sample

Sense by Sampling

sample too much data!

Sense then Compress

compress

decompress

sample

JPEGJPEG2000

…

Sparsity

pixels largewaveletcoefficients(blue = 0)

Sparsity

pixels largewaveletcoefficients(blue = 0)

widebandsignalsamples

largeGabor (TF)coefficients

time

frequ

ency

• Sparse signal: only K out of N coordinates nonzero

Concise Signal Structure

sorted index

• Sparse signal: only K out of N coordinates nonzero

– model: union of K-dimensional subspacesaligned w/ coordinate axes

Concise Signal Structure

sorted index

• Sparse signal: only K out of N coordinates nonzero

– model: union of K-dimensional subspaces

• Compressible signal: sorted coordinates decay rapidly with power-law

sorted index

power-lawdecay

Concise Signal Structure

What’s Wrong with this Picture?• Why go to all the work to acquire

N samples only to discard all but K pieces of data?

compress

decompress

sample

What’s Wrong with this Picture?linear processinglinear signal model(bandlimited

subspace)

compress

decompress

sample

nonlinear processingnonlinear signal model(union of subspaces)

Compressive Sensing• Directly acquire “compressed” data

via dimensionality reduction• Replace samples by more general

“measurements”

compressive sensing

recover

• Signal is -sparse in basis/dictionary– WLOG assume sparse in space domain

Sampling

sparsesignal

nonzeroentries

• Signal is -sparse in basis/dictionary– WLOG assume sparse in space domain

• Sampling

sparsesignal

nonzeroentries

measurements

Sampling

Compressive Sampling• When data is sparse/compressible, can

directly acquire a condensed representation with no/little information loss through linear dimensionality reduction

measurements sparsesignal

nonzeroentries

How Can It Work?• Projection

not full rank…

… and so loses information in general

• Ex: Infinitely many ’s map to the same(null space)

How Can It Work?• Projection

not full rank…

… and so loses information in general

• But we are only interested in sparse vectors

columns

How Can It Work?• Projection

not full rank…

… and so loses information in general

• But we are only interested in sparse vectors

• is effectively MxK

columns

How Can It Work?• Projection

not full rank…

… and so loses information in general

• But we are only interested in sparse vectors

• Design so that each of its MxK submatrices are full rank (ideally close to orthobasis)– Restricted Isometry Property (RIP)

columns

Restricted Isometry Property (RIP)• Preserve the structure of sparse/compressible

signals

K-dim subspaces

Restricted Isometry Property (RIP)• “Stable embedding”• RIP of order 2K implies: for all K-sparse x1 and

x2

K-dim subspaces

RIP = Stable Embedding• An information preserving projection preserves

the geometry of the set of sparse signals

• RIP ensures that

How Can It Work?• Projection

not full rank…

… and so loses information in general

• Design so that each of its MxK submatrices are full rank (RIP)

• Unfortunately, a combinatorial, NP-complete design problem

columns

Insight from the 70’s [Kashin, Gluskin]

• Draw at random– iid Gaussian– iid Bernoulli

…

• Then has the RIP with high probability provided

columns

Randomized Sensing• Measurements = random linear

combinations of the entries of the signal

• No information loss for sparse vectors whpmeasurements sparse

signal

nonzeroentries

CS Signal Recovery

• Goal: Recover signal from measurements

• Problem: Randomprojection not full rank(ill-posed inverse problem)

• Solution: Exploit the sparse/compressiblegeometry of acquired signal

CS Signal Recovery• Random projection

not full rank

• Recovery problem:givenfind

• Null space

• Search in null space for the “best” according to some criterion– ex: least squares

(N-M)-dim hyperplaneat random angle

• Recovery: given(ill-posed inverse problem) find (sparse)

• Optimization:

• Closed-form solution:

Signal Recovery

• Recovery: given(ill-posed inverse problem) find (sparse)

• Optimization:

• Closed-form solution:

• Wrong answer!

Signal Recovery

• Recovery: given(ill-posed inverse problem) find (sparse)

• Optimization:

• Closed-form solution:

• Wrong answer!

Signal Recovery

• Recovery: given(ill-posed inverse problem) find (sparse)

• Optimization:

Signal Recovery

“find sparsest vectorin translated nullspace”

• Recovery: given(ill-posed inverse problem) find (sparse)

• Optimization:

• Correct!

• But NP-Complete alg

Signal Recovery

“find sparsest vectorin translated nullspace”

• Recovery: given(ill-posed inverse problem) find (sparse)

• Optimization:

• Convexify the optimization

Signal Recovery

Candes Romberg Tao Donoho

• Recovery: given(ill-posed inverse problem) find (sparse)

• Optimization:

• Convexify the optimization

• Correct!

• Polynomial time alg(linear programming)

Signal Recovery

Compressive Sensing

• Signal recovery via optimization [Candes, Romberg, Tao; Donoho]

random measurements

sparsesignal

nonzeroentries

Compressive Sensing

• Signal recovery via iterative greedy algorithm– (orthogonal) matching pursuit [Gilbert, Tropp]– iterated thresholding

[Nowak, Figueiredo; Kingsbury, Reeves; Daubechies, Defrise, De Mol; Blumensath, Davies; …]

– CoSaMP [Needell and Tropp]

random measurements

sparsesignal

nonzeroentries

Greedy recovery algorithm #1• Consider the following problem

• Suppose we wanted to minimize just the cost, then steepest gradient descent works as

• But, the new estimate is no longer K-sparse

Iterated Hard Thresholding

update signal estimate

prune signal estimate(best K-term approx)

update residual

Greedy recovery algorithm #2• Consider the following problem

• Can we recover the support ?

sparsesignal

1 sparse

Greedy recovery algorithm #2• Consider the following problem

• If then gives the support of x

• How to extend to K-sparse signals ?

sparsesignal

1 sparse

Greedy recovery algorithm #2

sparsesignal

K sparse

residue:

find atom:

Add atom to support:

Signal estimate (Least squares over support)

Orthogonal matching pursuit

update residual

Find atom with largest support

Update signal estimate

Specialized solvers• CoSAMP [Needell and Tropp, 2009]

• SPG_l1 [Friedlander, van der Berg, 2008]http://www.cs.ubc.ca/labs/scl/spgl1/

• FPC [Hale, Yin, and Zhang, 2007]http://www.caam.rice.edu/~optimization/L1/fpc/

• AMP [Donoho, Montanari and Maleki, 2010]

many many others, see dsp.rice.edu/csandhttps://sites.google.com/site/igorcarron2/cscodes

CS Hallmarks• Stable

– acquisition/recovery process is numerically stable

• Asymmetrical (most processing at decoder) – conventional: smart encoder, dumb decoder– CS: dumb encoder, smart decoder

• Democratic– each measurement carries the same amount of information– robust to measurement loss and quantization– “digital fountain” property

• Random measurements encrypted• Universal

– same random projections / hardware can be used forany sparse signal class (generic)

Universality• Random measurements can be used for

signals sparse in any basis

Universality• Random measurements can be used for

signals sparse in any basis

Universality• Random measurements can be used for

signals sparse in any basis

sparsecoefficient

vector

nonzeroentries

Summary: CS

• Compressive sensing– randomized dimensionality reduction– exploits signal sparsity information– integrates sensing, compression, processing

• Why it works: with high probability, random projections preserve

information in signals with concise

geometric structures– sparse signals– compressible signals

Summary: CS• Encoding: = random linear combinations

of the entries of

• Decoding: Recover from via optimization

measurements sparsesignal

nonzeroentries

Image/Video specific signal models

and recovery algorithms

Transform basis• Recall Universality: Random measurements can

be used for signals sparse in any basis

• DCT/FFT/Wavelets …– Fast transforms; very useful in large scale problems

Dictionary learning• For many signal classes (ex: videos, light-fields),

there are no obvious sparsifying transform basis

• Can we learn a sparsifying transform instead ?

• GOAL: Given training data learn a “dictionary” D, such that

are sparse.

Dictionary learning• GOAL: Given training data learn a “dictionary” D, such that

are sparse.

Dictionary learning

• Non-convex constraint

• Bilinear in D and S

Dictionary learning

• Biconvex in D and S– Given D, the optimization problem is convex in sk

– Given S, the optimization problem is a least squares problem

• K-SVD: Solve using alternate minimization techniques– Start with D = wavelet or DCT bases– Additional pruning steps to control size of the dictionary

Aharon et al., TSP 2006

Dictionary learning

• Pros– Ability to handle arbitrary domains

• Cons– Learning dictionaries can be computationally intensive

for high-dimensional problems; need for very large amount of data

– Recovery algorithms may suffer due to lack of fast transforms

Models on image gradients• Piecewise constant

images– Sparse image gradients

• Natural image statistics– Heavy tailed distributions

Total variation prior

• TV norm– Sparse-gradient promoting norm

• Formulation of recovery problem

Total variation prior

• Optimization problem– Convex– Often, works “better” than transform

basis methods

• Variants– 3D (video)– Anisotropic TV

• Code– TVAL3– Many many others (see dsp.rice/cs)

Beyond sparsityModel-based CS

Beyond Sparse Models • Sparse signal model captures

simplistic primary structure

wavelets:natural images

Gabor atoms:chirps/tones

pixels:background subtracted

images

Beyond Sparse Models • Sparse signal model captures

simplistic primary structure

• Modern compression/processing algorithms capture richer secondary coefficient structure

wavelets:natural images

Gabor atoms:chirps/tones

pixels:background subtracted

images

Sparse Signals• K-sparse signals comprise a particular set of

K-dim subspaces

Structured-Sparse Signals• A K-sparse signal model comprises a particular

(reduced) set of K-dim subspaces[Blumensath and Davies]

• Fewer subspaces <> relaxed RIP <> stable recovery using

fewer measurements M

Wavelet Sparse

• Typical of wavelet transformsof natural signals and images (piecewise smooth)

Tree-Sparse• Model: K-sparse coefficients

+ significant coefficients lie on a rooted subtree

• Typical of wavelet transformsof natural signals and images (piecewise smooth)

Wavelet Sparse• Model: K-sparse coefficients

+ significant coefficients lie on a rooted subtree

• RIP: stable embedding

K-planes

Tree-Sparse• Model: K-sparse coefficients

+ significant coefficients lie on a rooted subtree

• Tree-RIP: stable embedding [Blumensath and Davies]

K-planes

Tree-Sparse

• Model: K-sparse coefficients + significant coefficients

lie on a rooted subtree

• Tree-RIP: stable embedding [Blumensath and Davies]

• Recovery: inject tree-sparse approx into IHT/CoSaMP

Recall: Iterated Thresholding

update signal estimate

prune signal estimate(best K-term approx)

update residual

Iterated Model Thresholding

update signal estimate

prune signal estimate(best K-term model approx)

update residual

Tree-Sparse Signal Recovery

target signal CoSaMP, (RMSE=1.12)

Tree-sparse CoSaMP (RMSE=0.037)

signal lengthN=1024

random measurementsM=80

L1-minimization(RMSE=0.751)

Clustered Signals

target Ising-modelrecovery

CoSaMPrecovery

LP (FPC)recovery

• Probabilistic approach via graphical model

• Model clustering of significant pixels in space domain using Ising Markov Random Field

• Ising model approximation performed efficiently using graph cuts [Cevher, Duarte, Hegde, Baraniuk’08]

Part 2: Compressive sensingMotivation, theory, recovery

• Linear inverse problems

• Sensing visual signals

• Compressive sensing– Theory– Hallmark– Recovery algorithms

• Model-based compressive sensing– Models specific to visual signals